5.2.6 Exercises for Section 5.2
Exercise 5.2.1 : Suppose we wish to store an n× n boolean matrix (0 and 1 elements only). We could represent it by the bits themselves, or we could represent the matrix by listing the positions of the 1’s as pairs of integers, each integer requiring ⌈log2n⌉ bits. The former is suitable for dense matrices; the latter is suitable for sparse matrices. How sparse must the matrix be (i.e., what fraction of the elements should be 1’s) for the sparse representation to save space?
Exercise 5.2.2 : Using the method of Section 5.2.1, represent the transition matrices of the following graphs:
(a) Figure 5.4.
(b) Figure 5.7.
Exercise 5.2.3 : Using the method of Section 5.2.4, represent the transition matrices of the graph of Fig. 5.3, assuming blocks have side 2.
Exercise 5.2.4 : Consider a Web graph that is a chain, like Fig. 5.9, with n nodes. As a function of k, which you may assume divides n, describe the representation of the transition matrix for this graph, using the method of Section 5.2.4
5.3 Topic-Sensitive PageRank
There are several improvements we can make to PageRank. One, to be studied in this section, is that we can weight certain pages more heavily because of their topic. The mechanism for enforcing this weighting is to alter the way random surfers behave, having them prefer to land on a page that is known to cover the chosen topic. In the next section, we shall see how the topic-sensitive idea can also be applied to negate the effects of a new kind of spam, called “‘link spam,”
that has developed to try to fool the PageRank algorithm.
5.3.1 Motivation for Topic-Sensitive Page Rank
Different people have different interests, and sometimes distinct interests are expressed using the same term in a query. The canonical example is the search query jaguar, which might refer to the animal, the automobile, a version of the MAC operating system, or even an ancient game console. If a search engine can deduce that the user is interested in automobiles, for example, then it can do a better job of returning relevant pages to the user.
Ideally, each user would have a private PageRank vector that gives the importance of each page to that user. It is not feasible to store a vector of length many billions for each of a billion users, so we need to do something
simpler. The topic-sensitive PageRank approach creates one vector for each of some small number of topics, biasing the PageRank to favor pages of that topic.
We then endeavour to classify users according to the degree of their interest in each of the selected topics. While we surely lose some accuracy, the benefit is that we store only a short vector for each user, rather than an enormous vector for each user.
Example 5.9 : One useful topic set is the 16 top-level categories (sports, med-icine, etc.) of the Open Directory (DMOZ).6 We could create 16 PageRank vectors, one for each topic. If we could determine that the user is interested in one of these topics, perhaps by the content of the pages they have recently viewed, then we could use the PageRank vector for that topic when deciding on the ranking of pages. 2
5.3.2 Biased Random Walks
Suppose we have identified some pages that represent a topic such as “sports.”
To create a topic-sensitive PageRank for sports, we can arrange that the random surfers are introduced only to a random sports page, rather than to a random page of any kind. The consequence of this choice is that random surfers are likely to be at an identified sports page, or a page reachable along a short path from one of these known sports pages. Our intuition is that pages linked to by sports pages are themselves likely to be about sports. The pages they link to are also likely to be about sports, although the probability of being about sports surely decreases as the distance from an identified sports page increases.
The mathematical formulation for the iteration that yields topic-sensitive PageRank is similar to the equation we used for general PageRank. The only difference is how we add the new surfers. Suppose S is a set of integers consisting of the row/column numbers for the pages we have identified as belonging to a certain topic (called the teleport set). Let eS be a vector that has 1 in the components in S and 0 in other components. Then the topic-sensitive Page-Rank for S is the limit of the iteration
v′= βM v + (1− β)eS/|S|
Here, as usual, M is the transition matrix of the Web, and|S| is the size of set S.
Example 5.10 : Let us reconsider the original Web graph we used in Fig. 5.1, which we reproduce as Fig. 5.15. Suppose we use β = 0.8. Then the transition matrix for this graph, multiplied by β, is
βM =
6This directory, found at www.dmoz.org, is a collection of human-classified Web pages.
5.3. TOPIC-SENSITIVE PAGERANK 167
B A
C D
Figure 5.15: Repeat of example Web graph
Suppose that our topic is represented by the teleport set S ={B, D}. Then the vector (1− β)eS/|S| has 1/10 for its second and fourth components and 0 for the other two components. The reason is that 1− β = 1/5, the size of S is 2, and eS has 1 in the components for B and D and 0 in the components for A and C. Thus, the equation that must be iterated is
v′ =
Here are the first few iterations of this equation. We have also started with the surfers only at the pages in the teleport set. Although the initial distribution has no effect on the limit, it may help the computation to converge faster.
Notice that because of the concentration of surfers at B and D, these nodes get a higher PageRank than they did in Example 5.2. In that example, A was the node of highest PageRank. 2
5.3.3 Using Topic-Sensitive PageRank
In order to integrate topic-sensitive PageRank into a search engine, we must:
1. Decide on the topics for which we shall create specialized PageRank vec-tors.
2. Pick a teleport set for each of these topics, and use that set to compute the topic-sensitive PageRank vector for that topic.
3. Find a way of determining the topic or set of topics that are most relevant for a particular search query.
4. Use the PageRank vectors for that topic or topics in the ordering of the responses to the search query.
We have mentioned one way of selecting the topic set: use the top-level topics of the Open Directory. Other approaches are possible, but there is probably a need for human classification of at least some pages.
The third step is probably the trickiest, and several methods have been proposed. Some possibilities:
(a) Allow the user to select a topic from a menu.
(b) Infer the topic(s) by the words that appear in the Web pages recently searched by the user, or recent queries issued by the user. We need to discuss how one goes from a collection of words to a topic, and we shall do so in Section 5.3.4
(c) Infer the topic(s) by information about the user, e.g., their bookmarks or their stated interests on Facebook.
5.3.4 Inferring Topics from Words
The question of classifying documents by topic is a subject that has been studied for decades, and we shall not go into great detail here. Suffice it to say that topics are characterized by words that appear surprisingly often in documents on that topic. For example, neither fullback nor measles appear very often in documents on the Web. But fullback will appear far more often than average in pages about sports, and measles will appear far more often than average in pages about medicine.
If we examine the entire Web, or a large, random sample of the Web, we can get the background frequency of each word. Suppose we then go to a large sample of pages known to be about a certain topic, say the pages classified under sports by the Open Directory. Examine the frequencies of words in the sports sample, and identify the words that appear significantly more frequently in the sports sample than in the background. In making this judgment, we must be careful to avoid some extremely rare word that appears in the sports sample with relatively higher frequency. This word is probably a misspelling that happened to appear only in one or a few of the sports pages. Thus, we probably want to put a floor on the number of times a word appears, before it can be considered characteristic of a topic.
Once we have identified a large collection of words that appear much more frequently in the sports sample than in the background, and we do the same for all the topics on our list, we can examine other pages and classify them by topic. Here is a simple approach. Suppose that S1, S2, . . . , Sk are the sets of words that have been determined to be characteristic of each of the topics on
5.4. LINK SPAM 169